The test will consist of problems that are similar to some of the following problems:

1) The problems of class activity 6B (background: activity 6A, and the exercises of section 6.1 -- note that these have answers in the text -- and problems 1, 2 of section 6.1).

2) Problem 3 of section 6.1, page 404.

3) a) Explain clearly and precisely what the term factor (the noun) means, as if you were telling 4th or 5th graders about this concept. b) Describe a problem or activity that you could give to children to help them learn the concept of factor. Explain briefly how you think your problem or activity would help the children learn the concept.

4) a) Explain clearly and precisely what the term multiple means, as if you were telling 4th or 5th graders about this concept. b) Describe a problem or activity that you could give to children to help them learn the concept of multiple. Explain briefly how you think your problem or activity would help the children learn the concept.

5) The problems of class activities 6G (background: activity 6F, the exercises of section 6.3, and problems 1-3 of section 6.3).

6) a) Explain clearly and precisely what the term greatest common factor means, as if you were telling 4th or 5th graders about this concept. b) Describe a problem or activity that you could give to children to help them learn the concept of greatest common factor. Explain briefly how you think your problem or activity would help the children learn the concept.

7) a) Explain clearly and precisely what the term least common multiple means, as if you were telling 4th or 5th graders about this concept. b) Describe a problem or activity that you could give to children to help them learn the concept of least common multiple . Explain briefly how you think your problem or activity would help the children learn the concept.

8) This problem refers to the shapes that come in standard pattern tile sets used in schools. If you can fill a shape completely with green triangles, thick blue rhombuses, red trapezoids, and yellow hexagons, then it is not possible to fill this same shape with orange squares and/or thin white rhombuses in addition to the other shapes above. Discuss very briefly how this is related to the nature of certain numbers.

9) Using the example 2/3 + 3/4, show how to add fractions and explain clearly why the procedure for adding fractions makes sense.

10) Using the example 2/3 - 1/2, show how to subtract fractions and explain clearly why the procedure for subtracting fractions makes sense.

11) Briefly describe two different ways to subtract 5 1/3 - 2 3/4. In each case, give your answer as a mixed number.

12) Use examples to discuss how to get equivalent fractions numerically and how to think about equivalent fractions in terms of pictures. Relate the numerical and pictorial points of view.

13) Exercises 2-4 of section 3.3, page 164.

14) Problem 4 of section 3.3, page 167.

15) Suppose you are teaching fraction multiplication. Make the case to your students that multiplication means the same thing whether we are multiplying fractions or whole numbers. Use story problems for 2 x 3 and 1/2 x 1/3 to illustrate.

16) In order to understand fraction multiplication deeply, we must be able to work simultaneously with different wholes. Using the example 2/3 x 2/5, explain why this is so. Discuss what difficulties a child might encounter in making sense of 2/3 x 2/5, and discuss how you might help the child progress. (Assume that the child already understands the meaning of fractions.)

17) In oder to understand fraction multiplication deeply, we must understand how to divide a whole number of objects into equal parts, such as dividing 8 cookies equally among 3 people. Explain specifically and in detail where dividing a whole number of objects into equal parts is needed in understanding 2/3 x 2/5.

18) Exercises 1-4 of section 4.9, page 294, 295. Problems 1-3 of section 4.9, pages 296, 297.

19) Exercise 2 for section 5.3, page 355, class activity 5H.

20) Problems 3, 4, 5 of section 5.5, page 374, class activity 5K, 1, 2, 3.

21) Exercise 3 for section 4.11, page 304, problem 2a for section 4.11, page 307, class activity 4EE # 3, page 144 (of activities).

22) Explain clearly why we put the decimal point where we do when we multiply 2.4 x 1.7. You may use graph paper to aid your explanation.

23) Give two different ways to multiply 6 1/2 x 4 2/3.

24) The class activities in volume 1 on pages 97, 98, 103, #4 on page 107 (referring to the pictures on pages 108, 109).

25) Given a sequence of pictures like those in class activity 10A, write a formula for how many small circles or squares are in the Nth picture.

26) Class activity 10B.

27) What equation does the sequence of patterns on page 157 (page 159) of the volume 2 class activities illustrate? Explain why.

28) Class activity 10I, 1, 2.

29) Given a repeating pattern (such as those shown in the copies from Navigating Through Algebra), write three different mathematical questions you could ask children about the pattern.